Yoshio Komori

Rooted tree analysis of the order conditions of row-type scheme for stochastic differential equations

Numerical schemes for initial value problems of stochastic differential equations (SDEs) are considered so as to derive the order conditions of ROW-type schemes in the weak sense. Rooted tree analysis, the well-known useful technique for the counterpart of the ordinary differential equation case, is extended to be applicable to the SDE case. In our analysis, the roots are bi-colored corresponding to the ordinary and stochastic differential terms, whereas the vertices have four kinds of label corresponding to the terms derived from the ROW-schemes. The analysis brings a transparent way for the weak order conditions of the scheme. An example is given for illustration.

Some issues in discrete approximate solution for stochastic differential equations

Abstract An evaluation method for numerical schemes of stochastic differential equations is treated. Discussing the source of errors in the discrete numerical solution, we highlight the effect of pseudo-random numbers upon the numerical solution, and point out the significance of the independencies of the series of them required in the numerical schemes. To discriminate the stochastic and deterministic parts in the errors more clearly, we propose a new two-dimensional linear test equation of multiplicative type whose analytical solution can be obtained readily. Our results are illustrated through some numerical examples.

Strong first order S-ROCK methods for stochastic differential equations

Explicit stochastic Runge-Kutta (SRK) methods are constructed for non-commutative Ito and Stratonovich stochastic differential equations. Our aim is to derive explicit SRK schemes of strong order one, which are derivative free and have large stability regions. In the present paper, this will be achieved by embedding Chebyshev methods for ordinary differential equations in SRK methods proposed by Roszler (2010). In order to check their convergence order, stability properties and computational efficiency, some numerical experiments will be performed.

Properties of the Weibull cumulative exposure model

Abstract This article is aimed at the investigation of some properties of the Weibull cumulative exposure model on multiple-step step-stress accelerated life test data. Although the model includes a probabilistic idea of Miners rule in order to express the effect of cumulative damage in fatigue, our result shows that the application of only this is not sufficient to express degradation of specimens and the shape parameter must be larger than 1. For a random variable obeying the model, its average and standard deviation are investigated on a various sets of parameter values. In addition, a way of checking the validity of the model is illustrated through an example of the maximum likelihood estimation on an actual data set, which is about time to breakdown of cross-linked polyethylene-insulated cables.

PARAMETER ESTIMATION BASED ON GROUPED OR CONTINUOUS DATA FOR TRUNCATED EXPONENTIAL DISTRIBUTIONS

ABSTRACT Parameter estimation based on truncated data is dealt with; the data are assumed to obey truncated exponential distributions with a variety of truncation time—a 1 data are obtained by truncation time b 1, a 2 data are obtained by truncation time b 2 and so on, whereas the underlying distribution is the same exponential one. The purpose of the present paper is to give existence conditions of the maximum likelihood estimators (MLEs) and to show some properties of the MLEs in two cases: 1) the grouped and truncated data are given (that is, the data each express the number of the data value falling in a corresponding subinterval), 2) the continuous and truncated data are given.

Supplement: Efficient weak second order stochastic Runge-Kutta methods for non-commutative Stratonovich stochastic differential equations

This paper gives a modification of a class of stochastic Runge–Kutta methods proposed in a paper by Komori (2007). The slight modification can reduce the computational costs of the methods significantly.

An easy parameter estimation by the EM algorithm in the new up-and-down method

Flashover voltage estimation has been carried out conventionally with the up-and-down method by Dixon and Mood. Recently, Hirose and Kato proposed a new version of the method to change the way of analyzing data. Although the method has better properties than the conventional one, it requires solving maximum likelihood equations. In this paper we reduce the troublesome task (e.g. implementation and the selection of a proper initial value) attendant on the requirement by using the expectation-maximization (EM) algorithm that gives a useful iterative formula to solve the equations. The iterative formula almost always can give the solutions of the likelihood equations because of its excellent global convergence of the EM algorithm.

Explicit Stochastic Runge‐Kutta Methods with Large Stability Regions

Our aim is to derive explicit Runge‐Kutta schemes for Stratonovich stochastic differential equations with a multidimensional Wiener process, which are of weak order 2 and which have large stability regions. This has been achieved by the use of a technique in Chebyshev methods for ordinary differential equations. In this talk, large stability regions of our schemes will be shown. Concerning convergence order and stability properties, the schemes will be tested in numerical experiments.

Easy estimation by a new parameterization for the three-parameter lognormal distribution

A new parameterization and algorithm are proposed for seeking the primary relative maximum of the likelihood function in the three-parameter lognormal distribution. The parameterization yields the dimension reduction of the three-parameter estimation problem to a two-parameter estimation problem on the basis of an extended lognormal distribution. The algorithm provides the way of seeking the profile of an object function in the two-parameter estimation problem. It is simple and numerically stable because it is constructed on the basis of the bisection method. The profile clearly and easily shows whether a primary relative maximum exists or not, and also gives a primary relative maximum certainly if it exists.

Weak Second Order Explicit Exponential Runge--Kutta Methods for Stochastic Differential Equations

We propose new explicit exponential Runge--Kutta methods for the weak approximation of solutions of stiff Ito stochastic differential equations (SDEs). We also consider the use of exponential Runge--Kutta methods in combination with splitting methods. These methods have weak order 2 for multidimensional, noncommutative SDEs with a semilinear drift term, whereas they are of order 2 or 3 for semilinear ordinary differential equations. These methods are A-stable in the mean square sense for a scalar linear test equation whose drift and diffusion terms have complex coefficients. We carry out numerical experiments to compare the performance of these methods with an existing explicit stabilized method of weak order 2.